A. The Mitzvah:
Moshe Rabbeinu established for the Jewish people that they are to read from the Torah scroll on Shabbos, Yom Tov, Yom Kippur, Rosh Chodesh, Chol Hamoed, Monday’s and Thursday’s. [This reading is of Rabbinical status.] Ezra established for the Torah to also be read on Mincha of Shabbos.
Shabbos: The Sages did not establish what portions should be read on each Shabbos, and hence in previous times two different customs existed regarding this matter. The widespread custom even in previous times was to read one Parsha per week and arrange to finish the entire Chumash annually.There were communities, however, who were accustomed to follow a triennial cycle and hence finish the entire Chumash every three years. This was the custom of Jewry in Eretz Yisrael during Talmudic times. Each Parsha was thus split up to three parts, with each part being read on one week. This minority custom became extinct several generations ago and hence the custom amongst all Jewry dating back many generations is to read one Parsha per week and complete the reading of the entire Torah annually. Practically, the weekly Torah portion for each Shabbos has the full backing of all Minhagei Yisrael, Jewish customs, and it is hence forbidden to switch the Parshiyos and change the custom. Furthermore, the Parsha of each week has a special connection with the events and times of that week, and it is all arranged with Divine providence.
Yom Tov: Moshe Rabbeinu established for the Jewish people that on every Yom Tov one is to read the Torah portion dealing with that Holiday. The Sages of the Mishneh and Gemara chose the exact portion of the Torah that is read on each holiday.
Kerias Hatorah for Ben Chutz Laretz in Eretz Yisrael when Parshiyos are not aligned:
If one traveled from the diaspora to Eretz Yisrael in a week that Eretz Yisrael is one Parsha ahead of the Diaspora, as occurs in certain years starting from the Shabbos after Pesach or the Shabbos after Shavuos, then one is to try and find a Minyan of Bnei Chutz La’aretz which will read the weekly Parsha of the Diaspora. Alternatively, one can ask the Baal Korei to begin the reading of Kohen from the previous Parsha and have him read until Levi of the current Parsha. If neither of the above options are possible, then due to lack of choice he has lost the previous Parsha and is to join the reading of Bnei Eretz Yisrael.
Shnayim Mikra: If one traveled to Eretz Yisrael in a week that Eretz Yisrael is reading a different Parsha than the Diaspora one is to read the Shnayim Mikra of both Parshiyos, the one which he is now missing in the Diaspora and the one which he will now hear in Eretz Yisrael. [If he returns to the Diaspora after Shabbos he is not required to repeat the Shnayim Mikra of the Parsha that was read in Eretz Yisrael and is now being read in the Diaspora.] In the event that he finds a Minyan of Bnei Chutz La’aretz who will only be reading the Parsha of Chutz La’aretz then he is to do Shnayim Mikra of only the Parsha of Chutz Laaretz.
Kerias Hatorah for Ben Eretz Yisrael in Chutz Laretz when Parshiyos are not aligned:
If one traveled from Eretz Yisrael to the Diaspora in a week that Eretz Yisrael is one Parsha ahead of the Diaspora, as occurs in certain years starting from the Shabbos after Pesach or the Shabbos after Shavuos, and thus in the Diaspora they will repeat the reading that he heard in Eretz Yisrael, then he is obligated to hear the Parsha a second time.
Shnayim Mikra: If one traveled from Eretz Yisrael to the Diaspora in a week that the Diaspora is reading the Parsha that was read the previous Shabbos in Eretz Yisrael He is not required to repeat Shnayim Mikra of that Parsha, even though he is required to hear the reading of the Torah.
Men versus congregation: It is debated amongst the Poskim if Kerias Hatorah is an obligation upon the congregation or individual. Practically, every man is required to place effort to be present by the Torah reading and hear every single word.
Women: Women, however, are not obligated in hearing Kerias Hatorah.
The Torah reading may take place anytime throughout the day, until sunset. Nevertheless, initially, it is to be read during Davening, as established by the Sages.
No Minyan: One who could not hear the Torah reading with the Minyan is to read the Parsha to himself from within a Chumash.
 Admur 282:1; 488:5; M”A 135:1; Rambam Tefila 12:1 “Moshe”; Rif Megillah 4; Bava Kama 82a that the prophets in the times of Moshe established it, See Kesef Mishneh ibid; Yerushalmi Megillah 4:1 “Moshe”; Miseches Sofrim 10:1; Mishneh Megillah 31a regarding Yom Tov; See P”M 135 A”A 1
 P”M 135 A”A 1 that so is implied from Setimas Haposkim in 135 and 685; Tosafus Megillah 17b
Other opinions: Some Poskim rule that Keiras Hatorah on Shabbos is a Biblical obligation. [Bach 685 and that so is opinion of Rashi, brought and negated in P”M ibid]
 M”A 135:1
 See Bach 685 “However, Moshe did not establish the order of what should be read on Shabbos until Ezra came along.” See Rambam 13:1-2 that the weekly Torah portion read on Shabbos is based on custom and not law, although Ezra established that certain Parshiyos are to be read at certain times, as brought in the Michaber 428 and 685. See Piskeiy Teshuvos 135:1
Other opinions: Some Poskim write that Moshe established which Parsha is to be read each Shabbos for all the Parshiyos of the Torah. [Aruch Hashulchan 282:2; See Zohar Vayakhel p. 206b “It is forbidden to strop an area that Moshe did not stop and it is forbidden to read a different weeks Parsah.”; Shulchan Hatahor 135:4 that even the double Parshiyos are a tradition of Moshe from Sinai; Piskeiy Teshuvos 135:1]
 Rambam ibid
 Rambam ibid
 Megillah 29b
 Aruch Hashulchan ibid
 See Divrei Torah 9:93; Shlah Vayeishev; Rebbe in various Sichos
 Admur 584:7; Michaber 584:2
 Admur ibid; 282:1; M”A 135:1; 535; Rambam Tefila 13:8; Mishneh Megillah 31a
The hint in scripture: The above institution is hinted to in the verse “Vayidaber Moshe Es Moadeiy Hashem El Bnei Yisrael”, from which we learn that Moshe established that each Prasha of the Moadim is to be read at its proper time. [Admur ibid]
 Admur ibid; Megillah ibid
 See Chayeh Levi 4/26; Piskeiy Teshuvos 285/9
 Kinyan Torah 6/12; See Betzeil Hachachmah 1/2-8
 Ketzos Hashulchan 72 footnote 3; Piskeiy Teshuvos 285/9
 Such as one traveled to Eretz Yisrael the week after a two day Shavuos or Pesach which coincided with Shabbos, in which case Eretz Yisrael is one Parsha ahead in its reading.
 Lehoros Nasan 3/13; Piskeiy Teshuvos 285/9
 Ikarei Hadaat 22/3; Ketzos Hashulchan 72 footnote 3; Piskeiy Teshuvos 285/9
 This can occur when the 2nd day of Pesach or Shavuos falls on Shabbos in the Diaspora and hence no Parsha is read, while in Eretz Yisrael the regular weekly Parsha was read. If one travels that week to the Diaspora he will be hearing the same Parsha that he heard the previous Shabbos in Eretz Yisrael.
 This ruling is evident from Admur 285/9 who does not require one to read Shnayim Mikra of the Yomim Tovim sections prior to each Yom Tov being that it was already read or will be read in its related Shabbos portion. Hence the same logistics apply here and there is no need to repeat Shnayim Mikra.
 M”B 135:1; Ketzos Hashulchan 25:14
 Ketzos Hashuclahn 25:14